The following lemma bounds \(s_n(z)\)  and \(\bar{s}_n(z)\)  on a region containing \(\bar{\lambda }_1, \ldots , \bar{\lambda }_{i_0}\) . The first point is a standard result. See, for example, Lemma \(8.17\)  of [1]}. The second point relies on the location of \(\lambda _1\) .
